parallel lines in hyperbolic geometry

In hyperbolic geometry, there are two kinds of parallel lines. If two lines do not intersect within a model of hyperbolic geometry but they do intersect on its boundary, then the lines are called asymptotically parallel or hyperparallel. (Note that, in the upper half plane model, any two vertical rays are asymptotically parallel. Thus, for consistency, ∞ is considered to be part of the boundary.) Any other set of parallel lines is called disjointly parallel or ultraparallel.